The Concept of Nature — the Tarner Lectures, 1920

Motion is essentially a relation between some object of nature and the one timeless space of a time-system. An instantaneous space is static, being related to the static nature at an instant. In perception when we see things moving in an approximation to an instantaneous space, the future lines of motion as immediately perceived are rects which are never traversed. These approximate rects are composed of small events, namely approximate routes and event-particles, which are passed away before the moving objects reach them. Assuming that our forecasts of rectilinear motion are correct, these rects occupy the straight lines in timeless space which are traversed. Thus the rects are symbols in immediate sense-awareness of a future which can only be expressed in terms of timeless space.

We are now in a position to explore the fundamental character of perpendicularity. Consider the two timesystems a and jS, each with its own timeless space and its own family of instantaneous moments with their instantaneous spaces. Let M and N be respectively a moment of a and a moment of /3. In M there is the direction of ^ and in A^" there is the direction of a. But M and N, being moments of different time-systems, intersect in a level. Call this level A. Then A is an instantaneous plane in the instantaneous space of M and also in the instantaneous space of N. It is the locus of all the event-particles which lie both in M and in N.

In the instantaneous space of M the level A is perpendicular to the direction of /S in M, and in the instantaneous space of N the level A is perpendicular to the direction of a in N. This is the fundamental property which forms the definition of perpendicularity. The symmetry of perpendicularity is a particular instance of the symmetry of the mutual relations between two time-systems. We shall find in the next lecture that it is from this symmetry that the theory of congruence is deduced.

The theory of perpendicularity in the timeless space of any time-system a follows immediately from this theory of perpendicularity in each of its instantaneous spaces. Let p be any rect in the moment M of « and let A be a level in M which is perpendicular to p . The locus of those points of the space of a which intersect M in event-particles on p is the straight line r of space a, and the locus of those points of the space of a which intersect M in event-particles on A is the plane / of space a. Then the plane / is perpendicular to the line r.

In this way we have pointed out unique and definite properties in nature which correspond to perpendicularity. We shall find that this discovery of definite unique properties defining perpendicularity is of critical importance in the theory of congruence which is the topic for the next lecture.

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I regret that it has been necessary for me in this lecture to administer such a large dose of four-dimensional geometry. I do not apologise, because I am really not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are ; and it is useless to disguise the fact that ' what things are ' is often very difficult for our intellects to follow. It is a mere evasion of the ultimate problems to shirk such obstacles.

CHAPTER VI
CONGRUENCE

The aim of this lecture is to establish a theory of congruence. You must understand at once that congruence is a controversial question. It is the theory of measurement in space and in time. The question seems simple. In fact it is simple enough for a standard procedure to have been settled by act of parliament ; and devotion to metaphysical subtleties is almost the only crime which has never been imputed to any English parliament. But the procedure is one thing and its meaning is another.

First let us fix attention on the purely mathematical question. When the segment between two points A and B is congruent to that between the two points C and D, the quantitative measurements of the two segments are equal . The equality of the numerical measures and the congruence of the two segments are not always clearly discriminated, and are lumped together under the term equality. But the procedure of measurement presupposes congruence. For example, a yard measure is applied successively to measure two distances between two pairs of points on the floor of a room. It is of the essence of the procedure of measurement that the yard measure remains unaltered as it is transferred from one position to another. Some objects can palpably alter as they move — for example, an elastic thread; but a yard measure does not alter if made of the proper material. What is this but a judgment of congruence applied to the train of successive positions of the yard

CH.vi] CONGRUENCE 121

measure? We know that it does not alter because we judge it to be congruent to itself in various positions. In the case of the thread we can observe the loss of self-congruence. Thus immediate judgments of congruence are presupposed in measurement, and the process of measurement is merely a procedure to extend the recognition of congruence to cases where these immediate judgments are not available. Thus we cannot define congruence by measurement.

In modern expositions of the axioms of geometry certain conditions are laid down which the relation of congruence between segments is to satisfy. It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes — in fact, a complete theory of non-metrical geometry. We then enquire about congruence and lay down the set of conditions — or axioms as they are called — which this relation satisfies. It has then been proved that there are alternative relations which satisfy these conditions equally well and that there is nothing intrinsic in the theory of space to lead us to adopt any one of these relations in preference to any other as the relation of congruence which we adopt. In other words there are alternative metrical geometries which all exist by an equal right so far as the intrinsic theory of space is concerned.

Poincare, the great French mathematician, held that our actual choice among these geometries is guided purely by convention, and that the effect of a change of choice would be simply to alter our expression of the physical laws of nature. By 'convention' I understand Poincare to mean that there is nothing inherent in nature itself giving any peculiar rdle to one of thesep. 122 congruence relations, and that the choice of one particular relation is guided by the volitions of the mind at the other end of the sense-awareness. The principle of guidance is intellectual convenience and not natural fact.

This position has been misunderstood by many of Poincare's expositors. They have muddled it up with another question, namely that owing to the inexactitude of observation it is impossible to make an exact statement in the comparison of measures. It follows that a certain subset of closely allied congruence relations can be assigned of which each member equally well agrees with that statement of observed congruence when the statement is properly qualified with its limits of error.